Báo cáo hóa học: " Research Article Distributed Cooperation among Cognitive Radios with Complete and Incomplete Information" pptx

The basic idea is that a secondary user a cognitive unlicensed user is able to properly sense the spectrum conditions, and, to increase efficiency in spectrum utilization, it seeks to unde

Volume 2009, Article ID 905185, 13 pages

doi:10.1155/2009/905185

Research Article

Distributed Cooperation among Cognitive Radios with

Complete and Incomplete Information

Lorenza Giupponi and Christian Ibars

Centre Tecnologic de Telecomunicacions de Catalunya (CTTC), Av del Canal Olimpic, s/n, 08860 Castelldefels, Spain

Correspondence should be addressed to Lorenza Giupponi,lorenza.giupponi@cttc.es

Received 30 January 2009; Accepted 20 May 2009

Recommended by John Chapin

This paper proposes that secondary unlicensed users are allowed to opportunistically use the radio spectrum allocated to the primary licensed users, as long as they agree on facilitating the primary user communications by cooperating with them The proposal is characterized by feasibility since the half-duplex option is considered, and incomplete knowledge of channel state information can be assumed In particular, we consider two situations, where the users in the scenario have complete or incomplete knowledge of the surrounding environment In the first case, we make the hypothesis of the existence of a Common Control Channel (CCC) where users share this information In the second case, the hypothesis of the CCC is avoided, which improves the robustness and feasibility of the cognitive radio network To model these schemes we make use of theory of exact and Bayesian potential games We analyze the convergence properties of the proposed games, and we evaluate the outputs in terms of quality of service perceived by both primary and secondary users, showing that cooperation for cognitive radios is a promising framework and that the lack of complete information in the decision process only slightly reduces system performance

Copyright © 2009 L Giupponi and C Ibars This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Cognitive Radio is a new paradigm in wireless

communica-tions to enhance utilization of limited spectrum resources It

is defined as a radio able to utilize available side information,

in a decentralized fashion, in order to efficiently use the radio

spectrum left unused by licensed systems The basic idea is

that a secondary user (a cognitive unlicensed user) is able

to properly sense the spectrum conditions, and, to increase

efficiency in spectrum utilization, it seeks to underlay,

overlay, or interweave its signals with those of the primary

(licensed) users, without impacting their transmission [1]

The interweave paradigm was the original motivation for

cognitive radio and is based on the idea of opportunistic

communications In fact, numerous measurement

cam-paigns have demonstrated the existence of temporary

space-time frequency voids, referred to as spectrum holes, which

are not in constant use in both licensed and unlicensed bands

and which can be exploited by secondary users (SUs) for

their communications The underlay paradigm encompasses

techniques that allow secondary communications assuming

that they have knowledge of the interference caused by its transmitter to the receivers of the primary users (PUs) Specifically, the underlay paradigm mandates that concur-rent primary and secondary transmissions may occur as long

as the aggregated interference generated by the SUs is below some acceptable threshold The overlay paradigm allows the coexistence of simultaneous primary and secondary communications in the same frequency channel as long as the SUs somehow aid the PUs, for example, by means of advanced coding or cooperative techniques In particular, in

a cooperative scenario the SUs may decide to assign part of their power to their own secondary communications and the remaining power to relay the PUs transmission [2]

While the most important challenge of the interweave paradigm is that of spectrum sensing, in order to realize

a reliable detection of the PUs, the significant challenge

to face in the underlay paradigm is that of estimating the aggregated interference at the PUs receivers that is being caused by the opportunistic activity of multiple SUs In literature, the analysis of the underlay paradigm for cognitive radio has often been realized by making use of game theoretic

approaches where SUs are modeled as the players of a game.

In this context, they make decisions in their own self interest

by maximizing their utility function, while influenced by

the other players decisions Generally, the different

control-lable transmission parameters in the communication (e.g

transmission power, frequency channel, etc.) represent the

strategies that can be taken by the players, and a function of,

for example, the (Signal-to-Interference-and-Noise Ratio)

SINR or the throughput is the utility of the game [3,4] The

main drawback of this approach is that the maximization of

the game utility function represents an incentive to reduce

the interference at the PUs receiver, but not a guarantee

that the aggregated interference generated by the SUs is

maintained below a certain threshold, especially in scenarios

where the spatial reuse is most challenging, for example,

where PUs receivers are passive or where SUs transmitters

are very close to PUs receivers In this context, cooperation of

SUs and PUs (overlay approach) can significantly reduce the

interference at the PUs receivers In particular, we propose

a cognitive radio scenario where concurrent primary and

secondary communications are allowed by exploiting spatial

reuse as long as the SUs cooperate with the PUs by relaying

their messages We consider two different cooperation

tech-niques: decode and forward (D&F) and amplify and forward

(A&F) In the proposed system, decisions about channel

selection and power allocation are taken distributively by

the SUs according to the maximization of their individual

utility These decisions strongly depend on those made

by the other SUs, since the PUs performances are limited

by the aggregated interference generated by all the SUs

simultaneously transmitting in their band This is why the

performance is analyzed using game-theoretic tools, already

proven good at modeling interactions in decision processes

In particular, we define two games to model channel and

power allocation for cognitive radios, underlay and overlay,

which can be formulated as exact potential games converging

to a pure strategy Nash equilibrium solution [5], and

we compare the overlay to the underlay scheme to learn

advantages and drawbacks of the proposed approach

However, inherent in this approach, as in nearly all

previous efforts, is the hypothesis of complete channel state

information among SUs; that is, the wireless channel gains

are assumed to be common knowledge across all SUs

This hypothesis implies the implementation of a common

control channel (CCC) where the distributed SUs can share

the information about their wireless channel gains In

literature, the hypothesis of such a fixed control channel

in a cognitive radio context has often been rejected [6],

since it requires a static assignment of licensed spectrum

before deployment, which is basically against the very

philosophy of cognitive radio Additionally, this solution

increases cost and complexity, limits scalability in terms

of device and traffic density, and is not robust to, for

example, jamming attacks As a result, in an effort to model

a more reliable, low-complexity and realistic self-organized

cognitive radio system, in the second part of this paper we

include uncertainty in the considered scenario, and we do

not rely on the existence of a preassigned CCC To this end,

we propose a Bayesian Potential Game (BPG), converging

to a Bayesian Nash Equilibrium, to model decentralized joint power and channel allocation for cooperative SUs with incomplete information Simulation results will show that the more realistic hypothesis of incomplete information only slightly reduces performances of PUs and SUs, and that cooperation among SUs significantly improves performances

of both PUs and SUs and that the improvement provided

by the overlay scheme is higher as the SU is closer to the primary receiver This results in a remarkable reduction of primary outage probability, since outages will typically occur

in primary receivers with nearby SUs The outline of the paper is organized as follows.Section 2describes the system model.Section 3presents the game theoretic model for the underlay and overlay games with complete (Section 3.1) and incomplete information (Section 3.2) Section 4 describes the simulation scenario.Section 5discusses relevant simula-tion results Finally,Section 6summarizes the conclusion

2 System Model

The cognitive radio network we consider consists of

M receiving PUs pairs, and N

transmitting-receiving SUs pairs (Figure 1) In this paper we will indicate the transmission power levels of the PUs’ transmitters

as p P

i,i = 1, , M, and the transmission power levels

of the SUs’ transmitters as p S

j,j = 1, , N PUs and

SUs, both transmitters and receivers, are randomly and uniformly distributed in a circular coverage region of a primary network with radius Rmax Primary communica-tions can be characterized by a long distance between the transmitting and the receiving device, whereas secondary communications are in general characterized by short range The nodes are either fixed or moving slowly (slower than the convergence of the proposed algorithm) The SUs are in charge of sensing the channel conditions and of choosing a transmission scheme which does not disrupt the communication of the PUs In this paper we consider and compare two communication paradigms for cognitive radio: underlay and overlay According to the underlay paradigm,

an SU distributively selects the frequency channel and the transmission power level to maximize its throughput while

at the same time not causing harmful interference to the PUs On the other hand, based on the overlay paradigm, besides selecting the transmission power and the frequency channel, the SUs devote part of their transmission power for relaying the primary transmission As a result, the SU’s transmission power level is split in two parts: (1) a power level p S j ,j = 1, , N for its own transmission, and (2) a

cooperation power level p S j ,j = 1, , N for relaying the

PU’s message on the selected band, wherep S j = p S j +p S j  The cooperative scheme used by the SUs is shown inFigure 2 We assume that the PU transmission is divided into frames, and each frame further into slots Relays are assumed to operate

in half-duplex mode Therefore, each relay listens to the primary transmission during one slot and transmits during the next The relay will choose, as part of its strategy, whether

to listen during even or odd slots We define these two slot subsets asS1andS2, respectively The primary transmission

SUt j

SUr j

SUt j+1

SUr j+1

PUr i

SUt j−1

SUr j−1

PUt i

PU communication

SU communication

Channeli , PU pair i

Figure 1: Cognitive system architecture

Primary

SecondaryS1

SecondaryS2

Listen Trans Listen Trans.

Listen Trans Listen Trans Listen

t

Figure 2: Half-duplex relaying scheme for secondary users Each

user chooses one slot to listen to the primary and retransmits in the

following slot Secondary users choose in which slot to transmit as

a part of their strategy

is continuous, and it does not interrupt to facilitate the relay

operation of the SUs In addition, we consider two different

relaying techniques: D&F and A&F In the D&F case the relay

(secondary user) decodes the primary signal, regenerates it,

and retransmits it during the next time slot In the event

that the relay is unable to decode, then it remains silent In

the A&F case, the relay simply stores the input during one

slot, amplifies it, and retransmits it during the next This

technique has the advantage that the relay is not required

to decode the signal On the other hand, the relay amplifies

input noise and interference as well as the useful signal

The performance of one technique or another will be better

depending on the ability of the relays to decode the signal,

and on the level of noise and interference at their input

The reader is referred to [7,8] for a thorough performance

comparison

Notice that the overlay scheme proposed and evaluated

in this paper is substantially different from the

property-rights model presented in [2], where PUs own the spectral

resource and may decide to lease part of it to SUs in exchange

for cooperation In fact, our overlay model does not require

PUs to be aware of the presence and identity of SUs It does,

however, require the PUs to be able to decode the cooperative transmission scheme employed

We shall analyze the network performance in terms of SINR and outage probability of both PUs and SUs As for the notation, we indicate withh PP i j the link gain between a PU’s transmitteri and a PU’s receiver j, with h PS i j the link gain between a PU’s transmitteri and an SU’s receiver j, with h SP

i j

the link gain between a SU’s transmitteri and a PU’s receiver

j, and with h SS

i j the link gain between an SU’s transmitteri

and an SU’s receiverj Finally, σ2is the noise power (assumed

to be equal in each channel)

2.1 Signal-to-Interference-and-Noise Ratio In the following

we calculate the expressions for the SINR for the underlay and overlay cases Notice that, for the PUs’ transmission,

we will consider an (Frequency Division Multiplexing) FDM scheme, so that only one PU is active per frequency channel

In the underlay paradigm, the SINRγ PU,u i for a pairi of

PUs in a frequency channelc iis given by

P

i h PP ii

N

j =1p S j h SP ji f

c j,c i

 +σ2, i =1, , M, (1) where f is defined as

f

c i,c j



˙

=

⎧

⎨

⎩

1, ifc i = c j,

Additionally, the SINR for the SUs is given by

γ i SU,u = p S i h SS ii

N

j =1,j / = i p S

j h SS

ji f

c j,c i

 +σ2, i =1, , N. (3)

In (3) it is assumed that the primary signal is known either

at the secondary receiver or at the secondary transmitter

In the first case, the interference of the primary signal can

be eliminated at the secondary receiver through a successive decoding strategy In the second case, it can be eliminated through dirty paper coding

For the overlay paradigm the expression of the SINR of the PUs depends on the relaying technique used

2.1.1 D&F In the following we will use the notation sl ito refer to the slot subset chosen by SU i, and we define the

function f 

f 

sl i,sl j



˙

=

⎧

⎨

⎩

1, ifsl i = sl j,

In the D&F approach, the SU must be able to correctly decode the primary signal to relay it In order to do that, the SINR of the primary signal, from PUj at SU transmitter i,

which is given by

P

j hPS ji

σ2+N

k =1,k / = i p Sh SS

ki f (c k,c i)f (sl k,sl i), i =1, , N

(5)

must be above the sensitivity threshold,ρ In the equation,

we use h ji andhki to denote the channel gains to the SU

transmitter, rather than the SU receiver, of SU pair i We

define the function

f 

γ PS i > ρ

˙

=

⎧

⎨

⎩

1, ifγ PS

i > ρ,

If γ PS i > ρ, then the SU may relay the primary signal.

We assume that the SU uses an encoding strategy that

is able to contribute to the received SINR Since the PU

will continue to transmit information, the scheme must

implement a distributed space-time coding scheme In order

to be realistic in terms of implementation, we do not assume

that PU and SUs may transmit phase-synchronously (i.e., to

perform distributed beamforming); therefore, their received

power adds up in incoherently The description of a specific

distributed space-time coding scheme is beyond the scope

of this paper, and the reader is referred to [2,9, 10] and

references therein for specific designs The SINR of the PU

i will be time-varying on the two slot subsetsS1,S2 and is

given by

γ i PU,o(Si)

= p

P

i h PP ii +N

j =1p S j  h SP ji f

c j,c i



f 

sl j,Si



f 

γ PS j > ρ

N

j =1p S j  h SP ji f

c j,c i



f 

sl j,Si

 +σ2

i =1, , M.

,

(7)

As conservative choice, in our performance evaluation we

consider the minimum SINR in any of the two slot subsets,

as it normally dominates the error rate Notice that, unlike

the underlay approach, part of the SU power contributes to

increasing the SINR by increasing the useful signal power

at the receiver (cooperation power) The SINR of the SU is

given by



i h SS ii

N

j =1,j / = i p S 

j h SS

ji f

c j,c i



f 

sl j,sl i

 +σ2,

i =1, , N.

(8)

2.1.2 A&F In the A&F mode, the SU retransmits the analog

signal received during the previous time slot The received

signal at SUj is given by

r j = p P

i hPS

N

k =1,k / = j

p S

k hSS

k j f

c k,c j



f 

sl k,sl j

 +σ2,

i =1, , M, j =1, , N.

(9)

Define the useful signal fraction of the transmitted primary signal as

P

ih PS

i j

p P i hPS

k =1,k / = j p S kh SS

k j f

c k,c j



f 

sl k,sl j

 +σ2,

i =1, , M, j =1, , N

(10) and the noise amplification fraction as

I j =

N

k =1,k / = j p S kh SS

k j f

c k,c j



f 

sl k,sl j

 +σ2

p P

i hPS

k =1,k / = j p S

kh SS

k j f

c k,c j



f 

sl k,sl j

 +σ2,

i =1, , M, j =1, , N.

(11)

In A&F mode, it is not possible to implement a space-time coding scheme since the relay may not do any process-ing of its received signal Therefore, the relay retransmits the signal in the same format as it was received, which creates

an artificial multipath channel for the receiver We assume

that the PU is able to take advantage of this multipath effect using similar techniques as those employed in conventional multipath resulting from propagation effects of the wireless medium As for the D&F case, the SINR of the PU will be time-varying on the two slot subsets, and again we consider the minimum SINR in any of the two For slot setSi,

γ PU,o i (Si)= p

P

i h PP ii +N

j =1p S j  R j h SP ji f

c j,c i



f 

sl j,Si



N

j =1



p S j +p S j  I j



h SP ji f

c j,c i



f 

sl j,Si

 +σ2,

i =1, , M.

(12) Finally, the SINR of the SUs is given by

S 

i h SS ii

N

j =1,j / = i



p S j +p S j  I j



h SS ji f

c j,c i



f 

sl j,sl i

 +σ2,

i =1, , N.

(13)

It is worth noting that in all the SINR expressions, the power relay and interference terms are not supposed to add

up coherently This assumption relaxes the synchronization requirements of primary and secondary users

2.2 Outage Probability Outage probability is defined as the

probability that a user i perceives an SINR γ i < γ dB,

where the threshold is set according to the primary receiver sensitivity

3 Game Theoretic Model

Game theory constitutes a set of mathematical tools to analyze interactions in decision making processes In this

paper we model joint channel and transmission power

selection in a cognitive radio scenario as the output of a game

where the players are theN SUs, the strategies are the choice

of the transmission power and of the frequency channel,

and the utility is a function of, (1) the interference each

SU causes to the surrounding PUs and SUs simultaneously

operating in the same frequency channel, (2) the interference

each SU receives from the surrounding SUs simultaneously

operating in the same frequency channel, and (3), the

satisfaction of each SU The SUs are aware of the interference

they receive, but to evaluate the interference they cause

to the surrounding PUs and SUs, they need information

about the wireless channel gains of their neighbors To

retrieve this information, we consider two cases In the

first case, we foresee the existence of a CCC where all the

users in the scenario share their transmission information,

so that the decisions of the SUs are made with complete

information Much attention has recently been paid to this

kind of channels; some examples are the Cognitive Pilot

Channel (CPC) [11] proposed by the E2R2/E3 consortium

[12] or the radio enabler proposed by the P1900.4 Working

Group [13] In the second case, taking into account that

the hypothesis of the existence of a CCC has often been

rejected in the cognitive radio literature, we provide a more

realistic and feasible proposal by avoiding the need of the

CCC and assuming that the decisions of the SUs are made

with incomplete information In this section we introduce

two games modeling the underlay and the overlay games, for

both the cases of complete (seeSection 3.1) and incomplete

(seeSection 3.2) information

3.1 An Exact Potential Game Formulation: Underlay and

Overlay Games with Complete Information We model this

problem as a normal form game, which can be

mathe-matically defined as Γ = {N, {S i } i ∈ N,{u i } i ∈ N }, where N

is the finite set of players (i.e., the N SUs), and S i is the

set of strategies s i associated with playeri We define S =

×S i, i ∈ N as the strategy space and u i : S → R as the

set of utility functions that the players associate with their

strategies For each player i in game Γ, the utility function

u i is a function of s i, the strategy selected by player i and

of the current strategy profile of the other players, which

is usually indicated with s − i The players make decisions

in a decentralized fashion, and independently, but they are

influenced by the other players decisions In this context, we

are interested in searching an equilibrium point for the joint

power and channel selection problem of the SUs from which

no player has anything to gain by unilaterally deviating

This equilibrium point is known as Nash equilibrium In

the following we introduce two games, representative of the

underlay and overlay paradigms, and we formulate them as

Exact Potential games

3.1.1 Underlay Game The underlay game is defined as

follows

(i)N is the finite set of players, that is, the SUs.

(ii) The strategies for playeri ∈ N are

(a) a power levelp i Sin the set of power levelsP S =

(p S, , p S

m);

(c1, , c l)

These strategies can be combined into a composite strategys i =(p S i,c i)∈ S i

(iii) The utility of each playeri is defined as follows: u(s i,s − i)= −

M

j =1

p S

i h SP

i j f

c i,c j



−

N

j =1,j / = i

p S j h SS ji f

c j,c i



−

N

j =1,j / = i

p i S h SS i j f

c i,c j



+b log

1 +p i S h SS ii

.

(14)

The expression presented in (14) consists of five terms The first and the third terms account for the interference the useri

is causing to the PUs and SUs simultaneously operating in the same frequency channel The second term accounts for the interference received by playeri from the SUs simultaneously

transmitting in the same frequency channel Finally, the fourth term only depends on the strategy selected by player

i and provides an incentive for individual players to increase

their power levels It is in fact considered that the players’ satisfaction increases logarithmically with their transmission power We weight this term by a coefficient b to give it more or less importance than the other terms of the utility function

3.1.2 Overlay Game The overlay game is defined as follows

(i)N is the finite set of players, that is, the SUs.

(ii) The strategies for playeri ∈ N are

(a) a power levelp S

i in the set of power levelsP S =

(p S, , p S

m);

(b) the power level p i S  that the player devotes to its own transmissions, in the set of power levels

P S 

= (p S1, , p S 

q ), where q is the order of set

P S 

; (c) the cooperative power level p S i  that the player devotes to relaying a PU transmission and which is computed asp S i  = p S i − p S i  The set

of these power levels,P S 

, is the same asP S 

;

(c1, , c l)

(e) a slot subsetsl ifrom the two possible subsetsS1

(even) andS2(odd)

These strategies can be combined into a composite strategys i =(p S i,p S i ,p S i ,c i,sl i)∈ S i We defineS =

×S,i ∈N as the strategy space

(iii) The utility of each playeri is defined as follows:

u(s i,s − i)= −

M

j =1

p S i  h SP i j f

c i,c j



−

N

j =1,j / = i

p S 

j h SS

ji f

c j,c i



f 

sl j,sl i



−

N

j =1,j / = i

p i S  h SS i j f

c i,c j



f 

sl i,sl j



+b log

1 +p S i  h SS ii



+

M

j =1

p i S  h SP i j f

c i,c j



f 

γ PS i > ρ

.

(15)

The expression presented in (15) consists of five terms

The first and the third terms account for the interference

perceived by the PUs and by the other SUs inc ifrom player

i, which only consists of the power the user i devotes to

the secondary transmission (i.e., p S i ) In case of SUs, p S i 

only affects users active in ci and in the same slot subset

The second term accounts for the interference generated on

player i by the SUs active in channel c i and in the same

slot subset as player i, sl i The fourth term represents an

incentive for the individual players to increase the power

level devoted to their own communications We weight this

term by a coefficient b to give it more or less importance

than the other terms of the utility function Finally, the

last term is a positive contribution to the utility function

and accounts for the benefit provided to the PUs by the

relaying realized by the SUs This term is positively defined to

encourage SUs to cooperate with PUs in exchange for using

their frequency channel Note that the term f (γ PS

i > ρ)

in the last term, which takes value 1 if the condition is

satisfied and 0 otherwise, only applies to the D&F scheme

It determines if the relay is not able to decode, and then it

does not increase its utility by cooperating, as it is not able to

do so For the A&F scheme, the relay always cooperates, and

therefore the term f (γ i PS > ρ) is always 1.

3.1.3 Existence of a Nash Equilibrium In order to have good

convergence characteristics for the above described games,

some mathematical properties have to be imposed on the

utility functions In particular, certain classes of games have

shown to always converge to a Nash Equilibrium when a

best response adaptive strategy is applied An example of

them is the class of Exact Potential Games A game Γ =

{N, {S i } i ∈ N,{u i } i ∈ N } is an Exact Potential game if there

exists a function Pot : S → R such that, for all i ∈ N,

s i, s i ∈ S i,

Pot(s i,s − i)−Pot s i,s − i

= u(s i,s − i)− u s i,s − i

(16)

The function Pot is called Exact Potential Function of the

gameΓ The potential function reflects the change in utility

for any unilaterally deviating player As a result, if PotPot

is an exact potential function of the game Γ, and s ∗ ∈ {argmaxs ∈ SPot(s)}is a maximizer of the potential function, then s ∗ is a Nash equilibrium of the game In particular, the best reply dynamic converges to a Nash Equilibrium in

a finite number of steps, regardless of the order of play and the initial condition of the game, as long as only one player acts at each time step, and the acting player maximizes its utility function, given the most recent actions of the other players For the previously formulated underlay and overlay games, we can define two exact potential functions, Potu(S)

and Poto(S).

(i) Underlay game Potential function:

Potu(s i,s − i)=

N

i =1

⎛

⎝− M

j =1

p S i h SP i j f

c i,c j

⎞

⎠

+

N

i =1

⎛

⎝− a

M

j =1,j / = i

p S j h SS ji f

c j,c i



−(1− a)

N

j =1,j / = i

p S

i h SS

i j f

c i,c j

⎞

⎠

+

N

i =1

b log

1 +p S

i h SS ii



.

(17)

(ii) Overlay game Potential function:

Poto(s i,s − i)

=

N

i =1

⎛

⎝− M

j =1

p S 

i h SP

i j f

c i,c j

⎞

⎠

+

N

i =1

⎛

⎝− a

N

j =1,j / = i

p S j  h SS ji f

c j,c i



f 

sl j,sl i



−(1− a)

N

j =1,j / = i

p S i  h SS i j f

c i,c j



f 

sl i,sl j

⎞

⎠

+

N

i =1

b log

1 +p S 

i h SS ii



+

N

i =1

M

j =1

p i S  h SP i j f

c i,c j



f 

γ i PS > ρ

,

(18)

wherea < 1 The proof that the underlay and overlay games,

with utility functions defined in (14) and (15) and with the potential functions defined in (17) and (18), are exact potential games is given in the appendix

3.2 A Bayesian Potential Game Formulation: Underlay and Overlay Games with Incomplete Information In a more

real-istic and feasible scenario, we should not rely on the existence

of a CCC where SUs share their transmission information As

a result, we consider a situation where incomplete knowledge

is available at the decision making agents In this section

we model joint channel and transmission power selection

for cognitive radios with incomplete information as the

output of a Bayesian Potential game In particular, we

consider two games of incomplete information, the underlay

and overlay Each one of these games is defined as Γ =

{N, {S i } i ∈ N,{η i } i ∈ N+,{ f H i(η i)} i ∈ N,{u i } i ∈ N }where

(i)N is the finite set of players, that is, the SUs, and N+

is a finite set withN+ ⊇ N, and N+\ N is the set of

outside players (i.e., the PUs);

(ii) for every i ∈ N, S i is the set of strategies

of player i, which have already been introduced

in case of complete knowledge for the underlay

game in Section 3.1.1 and for the overlay game in

Section 3.1.2;

(iii) a game of incomplete information, with respect to

a game of complete information, is characterized by

the player’s type, which embodies any information

that is not common knowledge to all players and is

relevant to the players’ decision making This may

include the player’s utility function, his belief about

other player’s utility functions, and so forth For every

i ∈ N+,H iis the finite set of possible types of player

i, η i = (h SS

1i, , h SS

i −1i,h SS i+1i, , h SS

Ni) ∈ Hi, which includes the wireless channel gains of playeri Each

player is assumed to observe perfectly its type but is

unable to observe the types of its neighbors;

(iv) f H i(η i) is a probability distribution onH = ×H i,i =

1, , N, with the a priori probability density

func-tion (PDF) onH defining the wireless channel gain

PDF;

(v) for everyi ∈ N, u i:S ×H → R is the utility function

of playeri.

The utility functions for playeri, for the underlay and

overlay games with incomplete information, are very similar

to those defined in (14) and (15), but besides being functions

of player i’s chosen strategy s i ∈ S i and other players’

strategies (s − i), they are functions of player i’s realized

channel gains η i ∈ H i and other SUs and PUs’ channel

gains (i.e., η − i) In particular, for the underlay game with

incomplete information,

u s i,s − i;η i,η − i

= −

M

j =1

p S

i h SP

i j f

c i,c j



−

N

j =1,j / = i

p S j h SS ji f

c j,c i



−

N

j =1,j / = i

p S i h SS i j f

c i,c j



+b log

1 +p S i h SS ii

,

(19)

and for the overlay game with incomplete information,

u s i,s − i;η i,η − i

= −

M

j =1

p S 

i h SP

i j f

c i,c j



−

N

j =1,j / = i

p S j  h SS ji f

c j,c i



f 

sl j,sl i



−

N

j =1,j / = i

p S 

i h SS

i j f

c i,c j



f 

sl i,sl j



+b log

1 +p S 

i h SS ii



+

M

j =1

p S i  h SP i j f

c i,c j



f 

γ PS i > ρ

.

(20)

It can be easily demonstrated (see the appendix) that the games with utility functions defined in (19) and (20) are Bayesian Potential games, if the following Potential functions are considered, for the underlay (21) and overlay (22) games with incomplete information, respectively:

PotuB s i,s − i;η i,η − i

=

N

i =1

⎛

⎝− M

j =1

p S

i h SP

i j f

c i,c j

⎞

⎠

+

N

i =1

⎛

⎝− a

N

j =1,j / = i

p S j h SS ji f

c j,c i



−(1− a)

N

j =1,j / = i

p S i h SS i j f

c i,c j

⎞

⎠

+

N

i =1

b log

1 +p S

i h SS ii

 ,

(21)

PotoB s i,s − i;η i,η − i

=

N

i =1

⎛

⎝− M

j =1

p S 

i h SP

i j f

c i,c j

⎞

⎠

+

N

i =1

⎛

⎝− a

N

j =1,j / = i

p S j  h SS ji f

c j,c i



f 

sl j,sl i



−(1− a)

N

j =1,j / = i

p S i  h SS i j f

c i,c j



f 

sl i,sl j

⎞

⎠

+

N

i =1

b log

1 +p S 

i h SS ii



+

N

i =1

M

j =1

p S i  h SP i j f

c i,c j



f 

γ i PS > ρ

.

(22)

0.01

0.02

0.03

0.04

0.05

0.06

0.08

0.07

0.09

0.1

Wireless channel gain (dB)

Figure 3: Wireless channel gain PMF derived by discretizing the

wireless channel gain PDF

As for the game with complete information, we need to

find an equilibrium point from which no player has anything

to gain by unilaterally deviating In a Bayesian game, this

point is a Bayesian Nash equilibrium; that is, a Bayesian Nash

equilibrium is a Nash equilibrium of a Bayesian game In

particular, a strategy profiles ∗ = (s ∗1, , s ∗ N) is a Bayesian

Nash equilibrium ifs ∗ i (η i) solves (23), assuming that types of

different players are independent:

s ∗ i η i

∈arg max

s i ∈ S

η − i

f H η − i

u i s i,s − i;η i,η − i

As it is proven in [14], the existence of a Bayesian

Nash equilibrium is an immediate consequence of the Nash

existence theorem As a result, considering that the potential

games have shown to always converge to a Nash Equilibrium

when a best response adaptive strategy is applied, it can be

derived that for the Bayesian Potential gameΓ there exists a

Bayesian Nash equilibrium, which maximizes the expected

utility function defined in (23)

4 Simulation Scenario

The scenario considered to evaluate the proposed framework

consists of a circular area with radius Rmax=150 m With

respect to the strategy space, the set of power levels P S =

(p S, , p S

m) is defined asP S =(0, 5, 10, 15, 20) dBm, that is,

m = 5 On the other hand, the SUs can be scheduled over

l =4 available frequency channels, so that the set of channels

C = (c1, , c l) is defined asC = (1, 2, 3, 4) Each channel

is assumed to have a bandwidthB c =200 KHz We consider

M =4 PUs pairs, one pair for each frequency channel, andN

SUs pairs, which at simulation start are randomly distributed

over thel frequency channels The PUs pairs are randomly

located in the scenario Specifically, the maximum distance

between a PU transmitter and a PU receiver is randomly

selected depending on their random position in the coverage

area On the other hand, the maximum distance between a

0 1 2 3

Simulation frames

Figure 4: Convergence of SUs pairs–frequency channel

SU transmitter and receiver is 20 m We consider a wireless channel gain ofh ii =(10/d2

ii), whered iiis the distance from transmitteri to receiver i The transmission power of a PU is

43 dBm The minimum SINR for a user not to be in outage is

γ =3dB In order to define the PDF of the wireless channel gains, we proceed by simulations We discretize the random variableR representing the distance between two nodes, and

accordingly the possible values of wireless channel gains, intoK equally spaced values In this way we generate a path

loss probability mass function (PMF) of the wireless channel gains, which is represented inFigure 3

5 Discussion

In this section we present simulation results to evaluate the performances of the proposed joint power and channel allocation algorithm for underlay and overlay approaches in both cases of complete and incomplete information First of all, we illustrate the convergence properties of the proposed algorithms The convergence of action updates in the overlay game for the case ofN =8 SUs in the scenario, and D&F relay mode, is shown in Figures4,5, and6 In particular,Figure 4

represents the choice of frequency channels, and Figures5

and 6 depict the selection of the transmission power, for the overlay game, which is split in two parts, the first one devoted to the secondary communication, and the second one to relaying the primary communication Notice how the players choose a variety of power levels and disperse, so as to transmit on a variety of frequency channels The convergence

of action updates of the underlay game is not shown since they are very similar to those of the overlay game Second,

Figure 7 compares the behavior of the Bayesian Potential Game with incomplete information (BPG) to the Exact Potential Game with complete information (EPG) It can be noticed how the lack of complete information only slightly reduces performances in terms of SINR for both PUs and SUs

In the following, we compare performance results of the underlay and overlay approaches, taking as a reference the D&F mode and the incomplete information case, since this

is the most feasible option.Figure 8compares performance

10

20

30

40

50

60

70

80

90

100

)

Simulation frames

Figure 5: Convergence of SUs pairs–transmission power devoted to

secondary communication

0

5

10

15

20

25

30

35

)

Simulation frames

Figure 6: Convergence of SUs pairs –transmission power devoted

to primary communication

results in terms of outage probability obtained by the

underlay and the overlay paradigms, as a function of the

number of SUs in the scenario It can be observed that

the overlay paradigm outperforms the underlay scheme in

terms of outage One of the reasons is that, in situations

characterized by the proximity of an SU transmitter to a

PU receiver, which are very critical for the underlay scheme,

the benefit gained by the cooperative approach increases In

fact, the message relayed by the SU is received with a higher

quality by the PU receiver Additionally, it is worth noting

that different results are obtained for different values of b

In particular, the lowerb, the more the SUs are discouraged

from increasing their transmission power at the expense of

the interference caused on the other users On the other

hand, Figure 9 compares SINR results for both PUs and

SUs It can be observed again that the overlay approach

benefits PUs but reduces the SUs performances, which is the

price to pay for being allowed to access primary channels

Let us now consider two different values of b for which

both the overlay and underlay games provide the PUs with

less than 3% of outage probability, (i.e., b = 10, for the

overlay game with incomplete information andb =0.001 for

the underlay game with incomplete information) It can be

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(dB) Primary user, PG-overlay Secondary user, PG-overlay Primary user, BPG-overlay Secondary user, BPG-overlay

Figure 7: SINR results: Bayesian Potential game with incomplete information versus Exact Potential game with complete informa-tion, for PUs and SUs

0 2 4 6 8 10 12 14 16 18

Number of SUs

b =3, underlay Bayesian

b =10, underlay Bayesian

b =3, overlay Bayesian

b =10, overlay Bayesian

Figure 8: PUs Outage probability for overlay and underlay games

observed fromFigure 10that even if the PUs results in terms

of outage are comparable, the SUs performances are reduced, when considering a lowerb, due to their lower transmission

power levels This demonstrates that, under the condition of limited interference on the PUs, also the SUs are benefited by cooperation In fact, they are allowed to transmit with higher power levels, as long as they devote a part of it for relaying primary communications; the results are more favorable to them than not cooperating and reducing theb parameter of

the game

Finally,Figure 11compares outage performances for the D&F and A&F relay modes, for the overlay game with

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(dB) Primary user, BPG-underlay

Secondary user, BPG-underlay

Primary user, BPG-overlay

Secondary user, BPG-overlay

Figure 9: SINR results: overlay versus underlay, for PUs and SUs

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(dB) Secondary user, BPG (b =0.001)-underlay

Secondary user, BPG (b =10)-overlay

Figure 10: SINR results for SUs considering different values of b:

underlay versus overlay, when the outage probability of PUs is 3%

incomplete information, when different values of detection

probability of the primary message at the SUs’ receivers are

considered It can be observed that when the SUs are able

to decode the PUs’ signals with a probability equal to 1, the

D&F relaying approach provides better performances than

the A&F scheme On the other hand, when the probability

of decoding the PU’s messages is reduced, it is also reduced

the probability that the SUs are able to cooperate with

the PUs Consequently, the A&F approach provides better

performances than the D&F

0 1 2 3 4 5 6 7 8

Number of SUs

A & F

D & F-detection probability=1

D & F-detection probability=0.8

Figure 11: Comparison of D&F and A&F outage performance results

6 Conclusion

In this paper we have introduced potential games to model joint channel and power allocation for cooperative and noncooperative cognitive radios Particular emphasis has been given to the feasibility of the proposed approach In fact, both the hypothesis of complete and incomplete information about the wireless channel gains is taken into account and compared, and the half-duplex option is considered for both D&F and A&F relay options of cooperative cognitive radios More in particular, we have proposed a cooperative scheme where SUs are allowed to use licensed channels as long as they provide compensation to PUs by means of cooperation (overlay approach), and we have compared it

to a scheme where cooperation between SUs and PUs is not considered (underlay approach) We have modeled these schemes by means of two Potential games, which are always characterized by a pure Nash equilibrium In addition to this, in order to avoid the implementation of a CCC, which would increase cost and complexity, we have considered the hypothesis of incomplete information, where SUs are unaware of the wireless channel gains of the other PUs and SUs Taking into account this additional hypothesis, both the underlay and overlay schemes have been modeled by means of Bayesian potential games converging to a pure Bayesian Nash equilibrium Simulation results have shown that cooperation benefits both PUs and SUs and that the hypothesis of incomplete information only slightly reduces performance results with respect to the case of complete information

Appendix

We prove that the game with the utility function defined in (20) and the potential function PotoB(S, H) defined in (22)

is a Bayesian potential game The same demonstration is also valid for the case of complete information with utility function (15) and potential function (18)